Complex K-AHSS for a closed oriented surface

By [A2] the page $E_2$ has nonzero entries $H^0,H^1,H^2$ in each even coefficient row, all free abelian, and vanishes in odd coefficient rows.

Every differential $d_r$ with $r\ge 2$ has target in the odd coefficient row $q-r+1$ when $r$ is even, and in the column $p+r>2$ when $r$ is odd; since the odd coefficient rows and the columns above the surface dimension $2$ vanish, all differentials $d_r$ for $r\ge 2$ vanish. The first differential is the cellular coboundary, so $E_2$ is already the cohomology page by construction.

The stable page has graded pieces $\mathbb{Z}$ in total degree zero for the rows contributing $H^0$ and $H^2$, and ${\mathbb{Z}}^{2g}$ in total degree one from $H^1$, so the associated graded of $K^0$ is ${\mathbb{Z}}^2$ and that of $K^1$ is ${\mathbb{Z}}^{2g}$.

Since all graded pieces are free, the finite filtrations split by [A3], so $K^0(S_g)\cong {\mathbb{Z}}^2\cong \mathbb{Z}\oplus {\tilde{K}}^0(S_g)$ with ${\tilde{K}}^0(S_g)\cong \mathbb{Z}$ and $K^1(S_g)\cong {\mathbb{Z}}^{2g}$; the extension data are not inferred from the collapse but from the splitting of free extensions.

This verifies the displayed groups $K^0\cong {\mathbb{Z}}^2$ and $K^1\cong {\mathbb{Z}}^{2g}$.